i = 0, which, with the Stern labor supply function, gives rise to a Tobit model. A more general fixed cost model assumes = b 0 + b 1 Z i + u 3i h min

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1 Economics 250a Lecture 4, Labor Supply Continued Outline 1. Simple labor supply models with corrections for non-participation (PS #2) 2. Other approaches to kinked budget sets - discretization 3. Dynamic labor supply - introduction and overview Readings Thomas A. Mroz. The Sensitivity of an Empirical Model of Married Women s Hours of Work to Economic and Statistical Assumptions. Econometrica, 55 (4) (July, 1987), pp Arthur van Soest. Structural Models of Family Labor Supply A Discrete Choice Approach. Journal of Human Resources 30(1), Winter David Card. Intertemporal Labor Supply: An Assessment. In Christopher Sims, editor, Advances in Econometrics, Sixth World Congresss (volume 2). Cambridge University Press, 1994 (nber version available on my web site) 1. Labor Supply and Corrections for Non-Participation We briefly discussed corner solutions in Lecture 2. We return to this issue using the setup in Mroz s study (which you will be replicating in the next problem set). We are trying to model (say) married women s labor supply, and we are prepared to take their partners earnings as exogenous. The setup includes a Stern labor supply function and a standard human capital (Mincer) model for log wages: h i = a 0 + a 1 log w i + a 2 y i + a 3 Z i + u 1i log w i = cz i + u 2i (1) where Z i is a vector of background variables (including characteristics of person i, the wife in a 2-person family, as well as the characteristics of her partner and her family). The parameters of interest are a 1 and a 2. Person i is observed working hours h i (or possibly h i + ɛ i, where ɛ i, is a measurement error) if h i h min i A pure reservation wage model assumes h min i = 0, which, with the Stern labor supply function, gives rise to a Tobit model. A more general fixed cost model assumes h min i = b 0 + b 1 Z i + u 3i which allows the lowest-hour job to depend on person-specific characteristics and a stochastic term. Working occurs if a 0 + a 1 log w i + a 2 y i + a 3 Z i + u 1i b 0 + b 1 Z i + u 3i 1

2 or ξ i = u 1i + a 1 u 2i u 3i (2) > (a 0 b 0 ) a 2 y i (a 3 + a 1 c b 1 )Z i. If we assume that (u 1i, u 2i, u 3i ) N(0, Σ) we get a very convenient setup. Equation (2) yields a simple probit model for participation with latent normal error ξ i. With 3 underlying error terms, this error is arbitrarily correlated with the error in the labor supply equation and with the error in the wage equation. (On the other hand, if u 3i = 0 we have only 2 independent errors). We can estimate both the wage model and the hours equation by 2-step Heckit, or IV-Heckit. The Heckit approach is a particular case of the so-called control function (CF) approach to censored data. The CF idea is to recognize that in the censored sample, the distribution of the error term is different for people with higher and lower probabilities of being in the sample. Heckman s original derivation assumed normality, which we will start with. A key fact is that if (z 1, z 2 ) N(0, Σ), the distribution of z 1 conditional on the event that z 2 > c, has a convenient functional form. Specifically, for joint normals we can always write: z 1 = r 1,2 z 2 + ν where r 1,2 cov[z 1, z 2 ]/var[z 2 ] = ρσ 1 / is the regression coefficient of z 1 on z 2 and ν is a normal variate with mean 0 that is independent of z 2. Thus E[z 1 z 2 > c] = r 1,2 E[z 2 z 2 > c] = r 1,2 E[ z 2 z 2 > c ] σ ( ) 2 φ c = ρσ 1 ( ) c ( ) = ρσ 1 1 Φ c ( ) φ c Φ ( c ) = ρσ 1 φ(φ 1 (p( c )) p( c ρσ 1 λ(p( c )) ) where p( c ) = Φ and we have used the result that for a standard normal variate ω, E(ω ω > a) = φ(a)/[1 Φ(a)]. Look at this carefully: it says that E[z 1 z 2 > c] = k λ(p( c )) where p is the probability of selection for the observation in question, k = ρσ 1 is a constant that has the same sign as ρ = correl(z 1, z 2 ) and λ(p) = φ(φ 1 (p)) p is a known, decreasing 2

3 function of p, with λ(p) > 0, λ(p) as p 0, and λ(p) 0 as p 1 (see the Figure at the end of the lecture). In our application z 2 > c is the condition for working (with probability p) and we will have 2 z 1 s: hours and wages, neither of which is observed for non-workers. In a 2-stage Heckit you first estimate the parameters of a reduced form probit model that determines whether the outcome (or outcomes) of interest are observed. You then take the parameters of this model, use them to predict p for each observation in the observed data set, and form λ(p). In the observed sample and E[h i w i, y i, Z i, working] = a 0 + a 1 log w i + a 2 y i + a 3 Z i +E[u 1i ξ i > (a 0 b 0 ) a 2 y i (a 3 + a 1 c b 1 )Z i ] h i = a 0 + a 1 log w i + a 2 y i + a 3 Z i + k 1 λ(p i ) + ν 1i E[log w i y i, Z i, working] = cz i + E[u 2i ξ i > (a 0 b 0 ) a 2 y i (a 3 + a 1 c b 1 )Z i ] log w i = cz i + k 2 λ(p i ) + ν 2i where (v i1, v 2i ) are good errors (orthogonal to the included covariates). If we are worried that log w i or y i are endogenous (or measured with a lot of error) then we have to apply IV-Heckit. Step 1 = probit for participation; step 2 = form λ(p) ; step 3: estimate first stage model for wages; step 4: estimate the labor supply model including λ(p) and instrumented values for wages. (Note: you should normally NOT do the 4th step yourself - use the 2SLS command). Another way to proceed would be to substitute log w i = cz i + u 2i into the hours equation, yielding h i = a 0 + (a 1 c + a 3 )Z i + a 2 y i + u 1i + a 1 u 2i ] which is only observed for participants. So we get a selection-corrected reduced form hours model h i = a 0 + (a 1 c + a 3 )Z i + a 2 y i + k 3 λ(p) + ν 3i We can estimate the parameters c using a selection-corrected wage model, and the parameters (a 1 c+a 3 ) and a 2 using a selection-corrected hours model. We can then do method of moments to infer a 1. An even more complex situation arises when we want to instrument y i (either because of measurement error, or a concern that some of the variables that drive y i also affect hours choices. This would lead to IV-Heckit with 2 endogenous variables Questions for discussion: 1) We clearly need instruments for the wage (given division bias). What are plausible candidates? 3

4 2) We also need variables that move λ(p) around so variables that determine the probability of participation but do not directly affect wages and hours. Thoughts? Non-normal alternatives: In general, for any pair of random variables (u, v) with some continuous density f(u, v) and marginal cdf F v (v): E[u v > c] = = = c c uf(u, v)dvdu f(u, v)dvdu ˆ 1 (ˆ uf(u) P (v > c) c ˆ 1 u P (v > c u)du P (v > c) = G(c) for some function G = G(Fv 1 (p)) = H(p) ) f(v u)dv du where p = P rob(v > c). So any two realizations of u from a censored sample where the censoring probabilities are the same have the same expected value. In general then, you can think of controlling for some flexible function of p in the censored regression model. If y = xβ + ɛ then E[y x, v > c] = xβ + H(p). for some H(p) = E[ɛ censoring event with prob. p]. complex models for the conditioning event. This idea can be generalized to more 2. Other approaches to kinked budget sets - discretization Kinked budget sets and non-participation can be handled in a unified way by thinking of the budget set as consisting of discrete points. For example, we could discretize hours choices into 1 hour bins, or 5 hour bins, or even 3 choices: h=0, h=20, h=40 (non-work, part-time, and full-time). This is arguably the state of the art for static labor supply. The simplest implementation is to pick a parametric utility function u(x, h; θ, z, ɛ), where θ represent parameters, z represent observable characteristics, and ɛ represents unobserved heterogeneity. Then assume some simple way of classifying observed choices into the discrete options {h 0 = 0, h 1, h 2,...h J }, and make a set of assumptions that allow you to associate an earnings amount {e 1, e 2,...e J } with each choice. If the worker faces a fixed wage w, for example, then you could assume e j = wh j. More generally, the earnings amounts e j can incorporate taxes and transfers, a premium or discount for part-time work, a fixed cost of working, etc). A worker with heterogeneity (z, ɛ) evaluates u(y + e j h j ; θ, z, ɛ) for each option and chooses the highest utility. In practice there are a lot of delicate issues. The most serious problem is that now we have to come up with values for (h j, e j ) for J choices, at most 1 of which is actually observed. 4

5 A second issue is how to incorporate random taste variation in the ɛ vector. point for this line of work is usually a multinomial logit model A starting u(y + e j, h j ; θ, z, ɛ) = v(y + e j, h j ; θ, z) + ɛ j where ɛ j are extreme value type 1 errors, and v is some convenient functional form. wage w was known for each person this would lead to a simple MNL choice model If the Pr(choice j w, z) = exp(v(y + e j(w), h j ; θ, z)) k v(y + e k(w), h k ; θ, z) A very simple example of this approach is as follows. First, assume that each person faces a parameteric gross market wage w, and that the hours choices are some fixed set, e.g., {0, 5, 10, 15...}. Knowing the tax schedule, find {e j (w)}, the net earnings at each h j for a person with gross wage w. Since the wage is not observed for non-workers (and may be measured with error), assume a DGP for wages that implies a density f(w z), and treat w as an endogenous outcome. Then the likelihood for the observed workers is of the form: Pr(choice j, w z) = Pr(choice j w, z) f(w z). The likelihood for non-workers is ˆ Pr(nonwork z) = Pr(choice j = 0 w, z) f(w z)dw An example of this approach is van Soest (1995). A more recent variant is: Arthur van Soest, Marcel Das and Xiaodong Gong, A Structural Labour Supply Model with Flexible Preferences. Journal of Econometrics 107 (2002), pp Intertemporal Labor Supply Intertemporal labor supply (ils) models are used to analyze responses to variation over time in wages or income. There are three main dimensions of intertemporal labor supply: (1) the pure lifecycle dimension. Wages have a hump-shaped pattern over the lifecycle. Do people respond to this variation? (2) the macro dimension. Hours of work vary over the business cycle. Can we interpret this as an endogenous response to wages? (3) the idiosyncratic dimension. A person may have temporarily higher wages in some period. How does he/she respond? Denote periods (e.g., years) by t. Consumption in period t is c t, hours of work are h t, the wage is w t. Individuals have an additively separable intertemporal utility function u(c t, h t ; a t ) + βu(c t+1, h t+1 ; a t+1 ) + β 2 u(c t+2, h t+2 ; a t+2 )

6 where a t is a preference shock in period t. The intertemporal budget contraint is A t+1 = (1 + r t )(A t + y t + w t h t c t ) where A t represent real assets in period t, and r t is the real interest rate from t to t + 1. Bellman equation is The The f.o.c. are Define V t (A t ) = max c t,h t u(c t, h t ; a t ) + βe t [V t+1 ((1 + r t )(A t + y t + w t h t c t ))] u c (c t, h t ; a t ) β(1 + r t )E t [V t+1(.)] = 0 u h (c t, h t ; a t ) + βw t (1 + r t )E t [V t+1(.)] = 0 λ t V t (A t ) = β(1 + r t )E t [V t+1(.)] = β(1 + r t )E t [λ t+1 ], where the second equality is from the envelope theorem and the third just uses the definition of λ t+1. We can rewrite the f.o.c. as u c (c t, h t ; a t ) = λ t u h (c t, h t ; a t ) = w t λ t implying a tangency condition (or within-period efficiency condition) u h (c t, h t ; a t ) u c (c t, h t ; a t ) = w t Note that the budget constraint within the period is c t = w t h t + y t [ A t r t A t ] So the within-period choices are the same as a person with the same utility function would make in a static problem with non-labor income y t S t, S t = A t r t A t The within-period f.o.c. define Frisch demands where c t = c F (w t, λ t, a t ) h t = h F (w t, λ t, a t ) 6

7 These are also called the lambda-constant or intertemporal consumption and hours choices. There are 4 elasticities log h F log hf log cf log cf,,, log w log λ log w log λ. The response of hours to wages holding constant λ is called the intertemporal substitution elasticity. Recall there are also compensated and uncompensated responses We showed in the first lecture that log h c (w, u) log h(w, y), log w log w log hf log w ( ) [ dc Ucc U = ch dh U hc U hh log hc (w,u) log w 0. Looking at the f.o.c. we get ] 1 [ 1 0 ] w λ So h F w = 1 U cc U hh U 2 ch U cc 0 λ U hc = λu cc The other derivatives are Using these relations we can show that h F λ c F λ c F w = λu ch = wu cc U hc = wu ch + U hh log h F log w log hf = log λ + c log c F wh log w log hf log w log hf So in the separable case (U ch = 0), we get that the = log λ. More generally if c wh this shows that hours have to vary more than consumption in response to expected wage variation. So an agent should increase savings when the wage is high. EXERCISE: Consider an agent with a given per-period utility function u(c, h) who solves a 1-period optimization: max u(c, h) s.t. c = wh + y c,h 7

8 Using Cramer s rule, find h w, h y. Show that h c w = λ w 2 U cc + 2wU ch + U hh and that 1 h c w 1 h f w 0. What is the intertemporal substitution elasticity? Start with the case where U(c, h) = φ(c) ψ(h). Then u c = λ t means that c t = c = const. (holding constant a t ). This was Friedman s assumption in the PIH model. In this case, as wages vary along an expected profile (so λ t = E t 1 [λ t ]) the agent finds an hours choice such that u h (c, h t ) u c (c, h t ) = w t This is illustrated in Figure 4.2 at the end. Note that utility U(c, h) = φ(c ) ψ(h t ) is lower in high wage-periods, but the person also gains savings,which have marginal value λ t = φ (c ) : so flow utility is φ(c ) ψ(h t ) + λ t (w t h t c ). In the non-separable case we have a locus of points (c, h) s.t. U c (c, h) = λ. As wages vary along an expected trajectory the agent finds a point on this locus with u h(c,h) u c(c,h) = w. Note that (ignoring preference changes over time): implying that U c (c F (w t, λ), h F (w t, λ)) = λ c F (w t, λ) w = U ch h F (w t, λ) U cc w and since hf (w t,λ) w > 0, and U cc < 0, the agent will consume more or less in high-wage periods depending on the sign of U ch. It is commonly asserted that U ch > 0. In this case the locus is negatively sloped, as shown in Figure 4.3. Dealing with Uncertainty 8

9 What happens if there are unexpected changes? Let s log-linearize the f.o.c.: log h t = a t + η log w t + δ log λ t And in the previous period Differencing we get But from the f.o.c recall so Now let log h t 1 = a t 1 + η log w t 1 + δ log λ t 1 log h t = log h t log h t 1 = a t + η log w t + δ(log λ t log λ t 1 ) λ t 1 = β(1 + r)e t 1 [λ t ] log λ t 1 = log[β(1 + r)] + log E t 1 [λ t ] log E t 1 [λ t ] = E t 1 log[λ t ] + φ t (see note ) and define the innovation in the log marginal utility of income as ξ t Combining all these terms we get So or log λ t = E t 1 log[λ t ] + ξ t log λ t 1 = log[β(1 + r)] + log E t 1 [λ t ] = log[β(1 + r)] + E t 1 log[λ t ] + φ t = log[β(1 + r)] + log λ t ξ t + φ t log λ t = log λ t 1 log[β(1 + r)] φ t + ξ t where: log h t == a t + η log w t + δξ t δ(φ t + log[β(1 + r)] This says that the change in hours depends on: the change in tastes a t the change in wages (with an elasticity η) the innovation in the log MU of income (with an elasticity δ 0) two terms which are constants An expected change in wages leaves λ t unchanged and leads to a change in hours η log w. An unexpected change, however, has wealth effects: an unexpected rise in w t will lower log λ t, especially to the extent it is expected to persist. Empirically this means that it is very 9

10 important to know whether wage innovations are mainly transitory or mainly persistent. The supply response to a transitory wage change is necessarily bigger (unless leisure is an inferior good). Aside what is the relationship of E[log x] and log E[x]? One easy case is when x is log-normally distributed: log x N(µ, ). (Wages are approximately log-normal so this is often a useful case for labor economics). In the log-normal case: E[x] = exp(µ σ2 ) so log E[x] = µ σ2 = E[log x] + 1 var[log x] 2 In the labor supply setting, then, if λ t was log-normal we would have that φ t = 1 2 var t 1[log λ t ] which could well vary over time or across people. 10

11 Normal Selection Correction Terms 3 Value of Correction Term linear approx (for p>.2) S= p Selected In Selected Out 2 linear approximation (for p<.8) S= 1.65p Probability of Selection

12 Figure 4.2: Intertemporal Labor Supply Choices: Separable Case Goods points with U L /U x = λ dissaving when w=w 1 Leisure saving when w=w 2 L=T

13 Figure 4.3: Intertemporal Labor Supply Choices: Non separable Case Goods points with U L /U x = λ dissaving when w=w 1 Leisure saving when w=w 2 L=T